#109873

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

435 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(2,7)(3,10)(4,5)(8,15)(11,13)(14,16)(17,18,22,26,23,27,21,35)(19,28,32,38,31,43,30,24)(20,25,33,40,34,41,36,39)(29,44,45,37,46,42,47,48);(1,2,5,9,6,7,4,12)(3,8,10,15,11,14,13,16)(17,19)(18,24,27,38)(20,29)(21,30)(22,32)(23,31)(25,37,41,48)(26,28,35,43)(33,45)(34,46)(36,47)(39,44,40,42);(1,3)(2,8)(4,13)(5,10)(6,11)(7,14)(9,15)(12,16)(17,20,22,33,23,34,21,36)(18,25,26,40,27,41,35,39)(19,29,32,45,31,46,30,47)(24,37,28,42,38,48,43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17069);